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Euler’s divergent series in arithmetic progressions. (English) Zbl 1443.11138
The author considers the series $$F(z)=\sum_{n=0}^\infty n!z^n$$ which converges for $$|z|_p\leq p^{1/(p-1)}$$ in the $$p$$-adic numbers $$\mathbb Q_p$$. The corresponding function is denoted by $$F_p(z)$$ for a fixed prime $$p$$. From this perspective, it makes sense to ask whether $$F_p(1)$$ is irrational or not. In the paper under review, the author proves that for a given rational number $$a/b$$ there exist infinitely many primes $$p$$ such that $$F_p(1)\neq a/b$$. Moreover, let $$m\geq 3$$. Then, the authors additionally show that infinitely many such primes $$p$$ are contained in only $$\varphi(m)/2$$ residue classes modulo $$m$$.

##### MSC:
 11J61 Approximation in non-Archimedean valuations
##### Keywords:
divergent series; global relation; $$p$$-adic number
Full Text:
##### References:
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